I'm looking for functions that are smooth ($C^\infty$) between $0 < x < \pi/2$ that satisfy the equation $$f(x)\, f(\pi/2-x) = 1$$ on the inteverval $0<x<\pi/2$. I know that the constant function $f=1$ satisfies this equation, as well as $f=\tan(x)$. Are these the only solutions?
It seems that there are infinitely many $C^\infty$ functions that work, so long as the power series at $x=\pi/4$ is consistent with the restrictions coming from taking derivatives of the above expression at $\pi/4$. Each of these power series should correspond to an analytic function that satisfies the above equation in a neighborhood of $x=\pi/4$. So the question seems to be how many analytic solutions are there to the above problem?